Lovejoy, JeremyJeremyLovejoyOsburn, RobertRobertOsburn2016-09-152016-09-152008 Oxfor2008-06-02Quarterly Journal of Mathematicshttp://hdl.handle.net/10197/7946In 1954, Atkin and Swinnerton-Dyer proved Dyson's conjectures on the rank of a partition by establishing formulae for the generating functions for rank differences in arithmetic progressions. In this paper, we prove formulae for the generating functions for rank differences for overpartitions. These are in terms of modular functions and generalized Lambert series.enThis article has been accepted for publication in Quarterly Journal of Mathematics ©: 2008 Oxford University Press. Published by Oxford University Press. All rights reserved.Elementary theory of partitionsPartitions of integersJournal Article59225727310.1093/qmath/ham0312016-08-10https://creativecommons.org/licenses/by-nc-nd/3.0/ie/